how can we draw 105 degree , 85 degree 110 degree angles using compass
For 105 degree angle
105 degree = 90 degree + 15 degree
So draw a 90 degree angle. Then draw a 60 degree angle with compass and bisect it twice so u get 60/2=30 30/2=15
90+15 = 105 degree
Construction of any angle with the use of only compass and scale is possible. However, this require a systematic construction procedures, which is explained below.
1) Initially construct a horizontal line.
2) Mark a point A on it; with A as center and any suitable radius draw a semi circular arc, which intersect the horizontal line at two distinct points, say P & Q.
3) With Q as center and with the same radius as in (2) above, cut an arc at R on the semi circle. Join A-R; So ∠QAR = 60 deg.
4) At A below PQ, draw an inclined line, preferably it subtends an acute angle with AQ. On this inclined line, mark 6 equal divisions using compass starting from A. Let these points be marked as 1,2,3,4,5 & 6. Join 6th point with Q. Then through the points 1,2,3,4 & 5 draw parallel lines to intersect AQ at 5 distinct points. Say, A₁, A₂, A₃, etc..
5) Through these points of intersection on AQ, draw lines parallel to AR to intersect the are QR at five different points, say Q₁, Q₂, Q₃, etc..
6) Join A-Q₁, A-Q₂, A-Q₃; ==> ∠QAQ₁ = 10 deg; ∠QAQ₂ = 20 deg, ∠QAQ₃ = 30 deg etc..
7) One arc length between successive points can be measured with compass, and any number of arcs from point R can be cut off. Every successive points make 10 deg. Thus for 110 deg, from R we need 5 successive divisions. Then join the 5th(say S) with A, making ∠QAS = 110 deg
8) Similar to the above any required angle can be marked.
9) For 85 deg, mark 80 and 90 angle. Using compass, bisect this arc, which will give 85 deg.